A262910 a(n) = Sum_{k=0..n} binomial(k+n-1,k)*binomial(k+n,2*k).

1, 2, 10, 59, 366, 2337, 15205, 100235, 667222, 4474733, 30188335, 204646532, 1392850785, 9511878729, 65144238981, 447263887479, 3077459618886, 21215286546705, 146500755609415, 1013180180867125, 7016536189029551, 48650933146617728, 337709155342663620

Offset: 0

Author

Vladimir Kruchinin, Oct 04 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1176

Crossrefs

Cf. A007863, A155112.

Programs

Maple

a := n -> hypergeom([-n, n, n+1], [1/2, 1], -1/4):
seq(round(evalf(a(n), 32)), n=0..21); # Peter Luschny, Oct 08 2015

Mathematica

Table[Sum[Binomial[k+n-1,k]*Binomial[k+n,2*k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Oct 04 2015 *)

Maxima

B(x):=sum(sum(binomial(i+n-1,i)*binomial(i+n,2*i+1),i,0,n-1)/n*x^n,n,1,30);
taylor(x*diff(B(x),x)/B(x),x,0,20);

PARI

a(n) = sum(k=0, n, binomial(k+n-1,k)*binomial(k+n,2*k));
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

Formula

G.f.: A(x) = x*B'(x)/B(x), where B(x)/x is g.f. of A007863.

Recurrence: 5*(n-1)*n*(35*n^2 - 143*n + 138)*a(n) = 2*(n-1)*(630*n^3 - 2889*n^2 + 3746*n - 1200)*a(n-1) - 2*(70*n^4 - 426*n^3 + 811*n^2 - 589*n + 150)*a(n-2) + 2*(n-3)*(2*n - 3)*(35*n^2 - 73*n + 30)*a(n-3). - Vaclav Kotesovec, Oct 04 2015

a(n) = hypergeom([-n, n, n+1], [1/2, 1], -1/4). - Peter Luschny, Oct 08 2015

a(n) = A155112(2n,n). - Alois P. Heinz, Sep 29 2022